If A is a set, a binary operation on A is a function B: Ax A→ A. Binary operations are ubiquitous in modern algebra, and their appearance there motivates the following notation: for (a1, a2) E A x A, we write aißa2 instead of the usual functional notation B(a₁, a2) for the image of (a1, a2) under 3. The most important (and motivating) instances of this (very general) notion for us are the addition and multiplication operations on a ring. 5. Let A be a set with a binary operation 3. An element e E A is an identity element for B if for all a € A, aße =a= eßa (for example, 0 € Z is an identity element for addition). Prove that, if an identity element for ß exists, then it is unique. (Hint: Suppose e, and e2 are both identity elements for ß. What can you say about e₁be₂?) 6. Let A be a set and let ß be a binary operation on A. We say that a subset BCA is closed under ß if for all a₁, a2 € B, a₁a₂ € B. (a) Prove that, if subsets B and C of A are closed under 6, then the intersection, BNC, is also closed under B. (b) For a € Z, define the set aZ = {ak ke Z}. Prove that aZ is closed under the addition operation on Z. (You can take for granted that the sum and product of two integers is an integer.)